On an ErdŐs–Kac-Type Conjecture of Elliott

IF 0.6 4区 数学 Q3 MATHEMATICS
Ofir Gorodetsky, Lasse Grimmelt
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引用次数: 0

Abstract

Elliott and Halberstam proved that $\sum_{p \lt n} 2^{\omega(n-p)}$ is asymptotic to $\phi(n)$. In analogy to the Erdős–Kac theorem, Elliott conjectured that if one restricts the summation to primes p such that $\omega(n-p)\le 2 \log \log n+\lambda(2\log \log n)^{1/2}$ then the sum will be asymptotic to $\phi(n)\int_{-\infty}^{\lambda} \mathrm{e}^{-t^2/2}\,\mathrm{d}t/\sqrt{2\pi}$. We show that this conjecture follows from the Bombieri–Vinogradov theorem. We further prove a related result involving Poisson–Dirichlet distribution, employing deeper lying level of distribution results of the primes.
论艾略特的一个埃尔德Ő斯-卡克型猜想
艾略特和哈尔伯斯塔姆证明了 $\sum_{p \lt n} 2^{\omega(n-p)}$ 是渐近于 $\phi(n)$ 的。与厄尔多斯-卡克定理类似、埃利奥特猜想,如果把求和限制在素数 p 上,使得 $\omega(n-p)\le 2 \log \log n+\lambda(2\log \log n)^{1/2}$ 那么和将渐近于 $\phi(n)\int_{-\infty}^{\lambda} \mathrm{e}^{-t^2/2}、\mathrm{d}t/\sqrt{2\pi}$.我们证明了这一猜想源于 Bombieri-Vinogradov 定理。我们进一步证明了涉及泊松-德里克利特分布的相关结果,运用了更深层次的素数分布结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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